In recent times, immersed/embedded/unfitted methods have seen a great development from the seminal ideas of Peskin [1]}. The key ideas in embedded methods is the use of grids that are not body-fitted, in which the geometry of the shapes to be simulated is immersed by way of computational geometry techniques.
In this work, we base the reduced order models on the Shifted Boundary Method (SBM), which is an embedded/unfitted finite element method originally proposed for the Poisson, Stokes and incompressible Navier-Stokes equations [2]}, [3]} and recently extended to wave equations and shallow water equations (SWE) [4]}.
In the SBM, a surrogate boundary is introduced in proximity of the true immersed boundary, and the boundary conditions are imposed on the surrogate boundary, with appropriate corrections that rely on Taylor expansions [2]}, [3]}.
The SBM does not require complicated data structures and numerical quadratures to integrate the governing equations on cut element, typical of cutFEM/XFEM approaches.
Compared to other embedded finite element methods such as XFEM or cutFEM [7]}, [8]}, [9]}, the SBM has also the advantage that the degrees of freedom (unknowns) stay the same for varying geometries, hence existing reduced order model methodologies are more easily adapted. Specifically, the total number of unknowns in SBM is determined by the background mesh and it is independent of the location of the embedded geometry.
In contrast, XFEM and other enriched finite element methods alike introduce new degrees of freedom such as the Heaviside functions within cut elements, hence the total number of unknowns typically varies with the embedded geometry locations and/or depends on a computationally expensive cutting of elements procedure.
